Advertisement

Continuum Mechanics and Thermodynamics

, Volume 29, Issue 6, pp 1241–1248 | Cite as

Finite-speed heat propagation as a consequence of microstructural changes

  • Paolo Maria Mariano
Original Article

Abstract

We show how a general description of microstructural changes in a macroscopically rigid conductor implies finite-speed propagation of temperature variations. In this way, we interpret once again Fourier’s paradox as a result of an insufficient representation of the structure of matter. The result is independent of the type of the material microstructure, provided that its changes are influenced by temperature variations. With the present treatment, we indicate a possible view on an old problem, already analyzed from different perspectives.

Keywords

Heat propagation Temperature finite speed Complex materials 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Both, S., Czél, B., Fülop, T., Gróf, Gy, Gyenis, Á., Kovács, C., Ván, P., Verhás, J.: Deviation from Fourier law in room-temperature heat pulse experiments. J. Non-Equilib. Thermodyn. 41, 41–48 (2015)Google Scholar
  2. 2.
    Capriz, G.: Continua with latent microstructure. Arch. Ration. Mech. Anal. 90, 43–56 (1985)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cattaneo, C.: Sulla conduzione del calore. Atti Sem. Mat. Fis. Univ. Modena 3, 83–101 (1948)MathSciNetMATHGoogle Scholar
  4. 4.
    Cimmelli, V.A.: Different thermodynamic theories and different heat conduction laws. J. Non-Equilib. Thermodyn. 34, 299–333 (2009)ADSCrossRefMATHGoogle Scholar
  5. 5.
    Dunn, J.E., Serrin, J.: On the thermomechanics of intertistitial working. Arch. Ration. Mech. Anal. 88, 95–133 (1985)CrossRefMATHGoogle Scholar
  6. 6.
    Ekoue, F., Fouache d’Halloy, A., Gigon, D., Plantamp, G., Zajdman, E.: Maxwell–Cattaneo regularization of heat equation. Int. J. Math. Comp. Phys. Electr. Comp. Eng. 7, 772–775 (2013)Google Scholar
  7. 7.
    Focardi, M., Mariano, P.M., Spadaro, E.N.: Multi-value microstructural descriptors for complex materials: analysis of ground states. Arch. Ration. Mech. Anal. 215, 899–933 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Frankel, J.I., Vick, B., Ösizik, M.N.: Flux formulation of hyperbolic heat conduction. J. Appl. Phys. 58, 3340–3345 (1985)ADSCrossRefGoogle Scholar
  9. 9.
    Giovine, P.: Nonclassical thermomechanics of granular materials. Math. Phys. Anal. Geom. 2, 179–196 (1999)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kovács, R., Ván, P.: Generalized heat conduction in heat pulse experiments. Int. J. Heat Mass Transfer 83, 613–620 (2015)CrossRefGoogle Scholar
  11. 11.
    Mariano, P.M.: Multifield theories in mechanics of solids. Adv. Appl. Mech. 38, 1–93 (2002)CrossRefGoogle Scholar
  12. 12.
    Mariano, P.M.: Mechanics of material mutations. Adv. Appl. Mech. 47, 1–91 (2014)CrossRefGoogle Scholar
  13. 13.
    Mariano, P.M., Modica, G.: Ground states in complex bodies. ESAIM Control Optim. Calc. Var. 15, 377–402 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Maxwell, J.C.: On the dynamic theory of gases. Philos. Trans. R. Soc. 157, 49–88 (1867)CrossRefGoogle Scholar
  15. 15.
    Mermin, N.D.: The topological theory of defects in ordered media. Rev. Mod. Phys. 51, 591–648 (1979)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Müller, I.: On the entropy inequality. Arch. Ration. Mech. Anal. 26, 118–141 (1967)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Müller, I.: The coldness, a universal function in thermoelastic bodies. Arch. Ration. Mech. Anal. 41, 319–332 (1974)MathSciNetMATHGoogle Scholar
  18. 18.
    Müller, I., Ruggeri, T.: Rational Extended Thermodynamics. Springer, New York (1998)CrossRefMATHGoogle Scholar
  19. 19.
    Ruggeri, T., Sugiyama, M.: Rational Extended Thermodynamics Beyond the Monoatomic Gas. Springer, Berlin (2015)CrossRefMATHGoogle Scholar
  20. 20.
    Straugham, B.: Heat Waves. Springer, New York (2011)CrossRefGoogle Scholar
  21. 21.
    Taitel, Y.: On the parabolic, hyperbolic and discrete formulation of the heat conduction equation. Int. J. Heat Mass Transfer 15, 369–371 (1972)CrossRefGoogle Scholar
  22. 22.
    Ván, P., Fülöp, T.: Universality in heat conduction theory—weakly nonlocal thermodynamics. Ann. Phys. 524, 470–478 (2012)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Vernotte, P.: Les paradoxes de la théorie continue de l’ équation de la chaleur. C. R. Acad. Sci. Paris 246, 3154–3155 (1958)MathSciNetMATHGoogle Scholar
  24. 24.
    Weymann, H.D.: Finite speed of propagation in heat conduction, diffusion and viscous shear motion. Am. J. Phys. 35, 488–496 (1967)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.DICeAUniversità di FirenzeFlorenceItaly

Personalised recommendations